The variable xt is known as the state variable. In Section 2 we recall some basics of geometric control theory as vector elds, Lie bracket and con-trollability. 0000003103 00000 n 0000051261 00000 n 12 0 obj h�T��n� E{�bʬR��&Ja�ly(vҳ0v�ր0.����(�3ý\�������=z�c��:q�k�g��.�X�v���U ����\$�;7zh[B?rsIq��a����E�Ѻ)����+W�5���0����^TxU3�*���De��� .Ai��M-c���h\$�3�����Σ�V����Y�+=�J�"{�\$;��Y~|��s�:��ĳo�>�\$e� �p0 �`@Ex1��Wsy�̝֦a' ~@�+Iy9�ߚ�A�4�L=Q��Nʠ}�(����&ud7/� �DqRў��K��q���{|c���. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. A Optimal Control Problem can accept constraint on the values of the control variable, for example one which constrains u(t) to be within a closed and compact set. It's a way of solving an optimization problem in continuous time. 9 0 obj %%EOF 0000010818 00000 n 0000002743 00000 n 0000018138 00000 n endobj 1. 0000004886 00000 n Although control theory has deep connections with classical areas of mathematics, such as the calculus of variations and the theory of differential equations, it did not become a field in its own right until the late 1950s and early 1960s. 0000004243 00000 n Let us begin to construct a mathematical model by setting x(t) = amount of output produced at time t≥ 0. 0000007283 00000 n optimal control theory. 0000009372 00000 n 0000001556 00000 n 0000041752 00000 n endobj 17 0 obj <> endobj A rigorous introduction to optimal control theory, with an emphasis on applications in economics. << /S /GoTo /D (section.5) >> 0000005279 00000 n In: Control Theory from the Geometric Viewpoint. << /S /GoTo /D (section.4) >> 0000060252 00000 n 0000013960 00000 n Initially, optimal control theory foundits application mainly in engi-neering disciplines like aeronautics, chemical and electrical engineering, robotics. These turn out to be sometimes subtle problems, as the following collection of examples illustrates. Agrachev A.A., Sachkov Y.L. 0000004640 00000 n endstream endobj 40 0 obj <> endobj 41 0 obj <>stream H��T�n�0��+t��DR%`͡�0��N���k4E�Y�~��m�,�I�-v0 ��{�㣜�o���aZ�͇�G2h�U��7���J���1���@���U�֤P�\�\���#O�I#���t�HqI�\���_m���q�Y�l�9�u��M{�_�� � 6H9Ѿp̕�e�|��\$��~YH[�����g�B��#2x�zuP@R�u8R{��{���7� 2�3�7�A�A����Yi�_4 ��m 0000003232 00000 n 24 0 obj As we proceed through the mathematical material, we will accompany each step with an economic example … << /S /GoTo /D (section.3) >> %���� Optimal control theory in economics. Introduction to Optimal Control 1.1 Some examples Example 1.1.1. 0000013509 00000 n Optimal control theory is a theory from mathematics.It looks at how to find a good (usually optimal) solution in a dynamic system. h�T�Mo�0��� 0000028901 00000 n dy dt g„x„t”,y„t”,t”∀t 2 »0,T… y„0” y0 This is a generic continuous time optimal control problem. An Economic Interpretation of Optimal Control Theory This section is based on Dorfman's (1969) excellent article of the same title. Economist 69a9. 28 0 obj endobj i��� �e���i ��Ub�c�������X#T���X��`�p�u� ���6��nBT�E�7��1V�>pn����W`�!��F۔ޤ-0��戮���aK�6�m����[\$~��^-��(��a`���L@l(ƶ� ��y� �nP In Section 3, that is the core of these notes, we introduce Optimal Control They each have the following form: max x„t”,y„t” ∫ T 0 F„x„t”,y„t”,t”dt s.t. 0000012914 00000 n 7�Z��P�C�����tH#D���ؔi������p4��ݍ��3t�iN��i�/��DB��y!�롶 |��6�8M7�n��fw���9{�A��]o.ޢ�痷������f��Z�"Q������7� ������dk��6�]'�2�.M��%)�5���]�����\$�*E���J>3!S�DJ/%R +U�I�X25�S�,f:�(O�4Ӗ���|�"�|N��ru��e[>����O�Lop}2v�a �~ YJ� (2004) Examples of Optimal Control Problems. Use features like bookmarks, note taking and highlighting while reading Optimal Control Theory and Static Optimization in Economics. /Length 1896 << /S /GoTo /D (section.2) >> endobj Pioneers and Examples. 0000017905 00000 n • then there is at least an optimal path for the state variable x ∗≡ {x 0,x1,...} The most common dynamic optimization problems in economics and ﬁnance have the following common assumptions • timing: the state variable xt is usually a stock and is measured at the beginning of period t and the control ut is usually a ﬂow and is measured 20 0 obj Rational behavior refers to a decision-making process that is based on making choices that result in an optimal level of benefit or utility. The curve of minimal length and the isoperimetric prob-lem Suppose we are interested to nd the curve of minimal length joining two distinct points in the plane. 0000005893 00000 n 32 0 obj << Many economic problems require the use of optimal control theory. 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